On regulated partitions

Abstract

This paper considers the combinatorics of continuous and Borel rectangular partitions of free actions of $\mathbb{Z}^n$ on $0$-dimensional Polish spaces, specifically the free part $F(2^{\mathbb{Z}^n})$ of the shift action of $\mathbb{Z}^n$ on the space $2^{\mathbb{Z}^n}$. This is done through the study of a corresponding notion of regulated partitions of $\mathbb{R}^n$. The main concepts studied are the continuous and Borel {\em regulation} numbers of the partition. This is defined as the maximum number of rectangles in the corresponding regulated partition that can intersect in a point. The continuous and Borel regulation numbers $γ_c$, $γ_B$ are the minimum possible values of these numbers as we range over continuous (respectively Borel) rectangular partitions of $F(2^{\mathbb{Z}^n})$. It is shown that for $n=2$ that $γ_c=γ_B=3$, and for $n \geq 3$ that $n+2\leq γ_B \leq γ_c \leq 3\cdot 2^{n-2}$. For $n=3$ we improve this to $γ_c=γ_B=5$. This shows a striking difference between the Borel combinatorics of dimension $n=2$ and dimensions $n>2$.

Figures (12)

• Figure 1: A decomposition of matrix $M$.
• Figure 2: A decomposition of $\bigcup\mathcal{A}$.
• Figure 3: A hypothetical concave maximal surface.
• Figure 4: In search for the next virtual maximal surface.
• Figure 5: The existence of the virtual maximal surface $M_2$.

Theorems & Definitions (101)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Definition 2.4
• Lemma 2.5
• proof
• Example 2.6